Theorem signslema 28159

Description: Computational part of signwlemn . (Contributed by Thierry Arnoux, 29-Sep-2018.)

Hypotheses

Ref	Expression
signslema.1	|- ( ph -> E e. NN0 )
signslema.2	|- ( ph -> F e. NN0 )
signslema.3	|- ( ph -> G e. NN0 )
signslema.4	|- ( ph -> H e. NN0 )
signslema.5	|- ( ph -> ( E < G /\ -. 2 || ( G - E ) ) )
signslema.6	|- ( ph -> ( ( H - G ) - ( F - E ) ) e. { 0 , 2 } )

Assertion

Ref	Expression
signslema	|- ( ph -> ( F < H /\ -. 2 || ( H - F ) ) )

Proof of Theorem signslema

Step	Hyp	Ref	Expression
1		signslema.5	. . . . . 6 |- ( ph -> ( E < G /\ -. 2 || ( G - E ) ) )
2	1	simpld 459	. . . . 5 |- ( ph -> E < G )
3	2	adantr 465	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> E < G )
4		signslema.4	. . . . . . . . . 10 |- ( ph -> H e. NN0 )
5	4	nn0cnd 10850	. . . . . . . . 9 |- ( ph -> H e. CC )
6		signslema.2	. . . . . . . . . 10 |- ( ph -> F e. NN0 )
7	6	nn0cnd 10850	. . . . . . . . 9 |- ( ph -> F e. CC )
8	5, 7	subcld 9926	. . . . . . . 8 |- ( ph -> ( H - F ) e. CC )
9		signslema.3	. . . . . . . . . 10 |- ( ph -> G e. NN0 )
10	9	nn0cnd 10850	. . . . . . . . 9 |- ( ph -> G e. CC )
11		signslema.1	. . . . . . . . . 10 |- ( ph -> E e. NN0 )
12	11	nn0cnd 10850	. . . . . . . . 9 |- ( ph -> E e. CC )
13	10, 12	subcld 9926	. . . . . . . 8 |- ( ph -> ( G - E ) e. CC )
14	8, 13	subeq0ad 9936	. . . . . . 7 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 0 <-> ( H - F ) = ( G - E ) ) )
15	14	biimpa 484	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( H - F ) = ( G - E ) )
16	15	breq2d 4459	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( 0 < ( H - F ) <-> 0 < ( G - E ) ) )
17	6	nn0red 10849	. . . . . . 7 |- ( ph -> F e. RR )
18	4	nn0red 10849	. . . . . . 7 |- ( ph -> H e. RR )
19	17, 18	posdifd 10135	. . . . . 6 |- ( ph -> ( F < H <-> 0 < ( H - F ) ) )
20	19	adantr 465	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( F < H <-> 0 < ( H - F ) ) )
21	11	nn0red 10849	. . . . . . 7 |- ( ph -> E e. RR )
22	9	nn0red 10849	. . . . . . 7 |- ( ph -> G e. RR )
23	21, 22	posdifd 10135	. . . . . 6 |- ( ph -> ( E < G <-> 0 < ( G - E ) ) )
24	23	adantr 465	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( E < G <-> 0 < ( G - E ) ) )
25	16, 20, 24	3bitr4rd 286	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( E < G <-> F < H ) )
26	3, 25	mpbid 210	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> F < H )
27		0re 9592	. . . . . 6 |- 0 e. RR
28	27	a1i 11	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 e. RR )
29	22, 21	resubcld 9983	. . . . . 6 |- ( ph -> ( G - E ) e. RR )
30	29	adantr 465	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( G - E ) e. RR )
31	18, 17	resubcld 9983	. . . . . 6 |- ( ph -> ( H - F ) e. RR )
32	31	adantr 465	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( H - F ) e. RR )
33	2	adantr 465	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> E < G )
34	23	adantr 465	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( E < G <-> 0 < ( G - E ) ) )
35	33, 34	mpbid 210	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 < ( G - E ) )
36		2pos 10623	. . . . . . . 8 |- 0 < 2
37		breq2 4451	. . . . . . . 8 |- ( ( ( H - F ) - ( G - E ) ) = 2 -> ( 0 < ( ( H - F ) - ( G - E ) ) <-> 0 < 2 ) )
38	36, 37	mpbiri 233	. . . . . . 7 |- ( ( ( H - F ) - ( G - E ) ) = 2 -> 0 < ( ( H - F ) - ( G - E ) ) )
39	38	adantl 466	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 < ( ( H - F ) - ( G - E ) ) )
40	29, 31	posdifd 10135	. . . . . . 7 |- ( ph -> ( ( G - E ) < ( H - F ) <-> 0 < ( ( H - F ) - ( G - E ) ) ) )
41	40	biimpar 485	. . . . . 6 |- ( ( ph /\ 0 < ( ( H - F ) - ( G - E ) ) ) -> ( G - E ) < ( H - F ) )
42	39, 41	syldan 470	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( G - E ) < ( H - F ) )
43	28, 30, 32, 35, 42	lttrd 9738	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 < ( H - F ) )
44	19	adantr 465	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( F < H <-> 0 < ( H - F ) ) )
45	43, 44	mpbird 232	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> F < H )
46	5, 10, 7, 12	sub4d 9975	. . . . 5 |- ( ph -> ( ( H - G ) - ( F - E ) ) = ( ( H - F ) - ( G - E ) ) )
47		signslema.6	. . . . 5 |- ( ph -> ( ( H - G ) - ( F - E ) ) e. { 0 , 2 } )
48	46, 47	eqeltrrd 2556	. . . 4 |- ( ph -> ( ( H - F ) - ( G - E ) ) e. { 0 , 2 } )
49		ovex 6307	. . . . 5 |- ( ( H - F ) - ( G - E ) ) e. _V
50	49	elpr 4045	. . . 4 |- ( ( ( H - F ) - ( G - E ) ) e. { 0 , 2 } <-> ( ( ( H - F ) - ( G - E ) ) = 0 \/ ( ( H - F ) - ( G - E ) ) = 2 ) )
51	48, 50	sylib 196	. . 3 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 0 \/ ( ( H - F ) - ( G - E ) ) = 2 ) )
52	26, 45, 51	mpjaodan 784	. 2 |- ( ph -> F < H )
53	1	simprd 463	. . . . 5 |- ( ph -> -. 2 || ( G - E ) )
54	53	adantr 465	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> -. 2 || ( G - E ) )
55	15	breq2d 4459	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( 2 || ( H - F ) <-> 2 || ( G - E ) ) )
56	54, 55	mtbird 301	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> -. 2 || ( H - F ) )
57		2z 10892	. . . . . . 7 |- 2 e. ZZ
58	9	nn0zd 10960	. . . . . . . 8 |- ( ph -> G e. ZZ )
59	11	nn0zd 10960	. . . . . . . 8 |- ( ph -> E e. ZZ )
60	58, 59	zsubcld 10967	. . . . . . 7 |- ( ph -> ( G - E ) e. ZZ )
61		dvdsaddr 13880	. . . . . . 7 |- ( ( 2 e. ZZ /\ ( G - E ) e. ZZ ) -> ( 2 || ( G - E ) <-> 2 || ( ( G - E ) + 2 ) ) )
62	57, 60, 61	sylancr 663	. . . . . 6 |- ( ph -> ( 2 || ( G - E ) <-> 2 || ( ( G - E ) + 2 ) ) )
63	53, 62	mtbid 300	. . . . 5 |- ( ph -> -. 2 || ( ( G - E ) + 2 ) )
64	63	adantr 465	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> -. 2 || ( ( G - E ) + 2 ) )
65		2cnd 10604	. . . . . . . 8 |- ( ph -> 2 e. CC )
66	8, 13, 65	subaddd 9944	. . . . . . 7 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 2 <-> ( ( G - E ) + 2 ) = ( H - F ) ) )
67	66	biimpa 484	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( ( G - E ) + 2 ) = ( H - F ) )
68	67	breq2d 4459	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( 2 || ( ( G - E ) + 2 ) <-> 2 || ( H - F ) ) )
69	68	notbid 294	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( -. 2 || ( ( G - E ) + 2 ) <-> -. 2 || ( H - F ) ) )
70	64, 69	mpbid 210	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> -. 2 || ( H - F ) )
71	56, 70, 51	mpjaodan 784	. 2 |- ( ph -> -. 2 || ( H - F ) )
72	52, 71	jca 532	1 |- ( ph -> ( F < H /\ -. 2 || ( H - F ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   {cpr 4029   class class class wbr 4447  (class class class)co 6282   RRcr 9487   0cc0 9488   + caddc 9491   < clt 9624   - cmin 9801   2c2 10581   NN0cn0 10791   ZZcz 10860   || cdivides 13843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-dvds 13844

This theorem is referenced by: (None)